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Let $\frak{g}$ be a semisimple Lie algebra, and let $({-},{-})$ be an invariant inner product on $\frak{g}$. The Chevalley–Eilenberg complex $C^*(\frak{g})$ has a natural Poisson bracket of degree $-2$; if we think of the generators of the exterior algebra $C^*(\frak{g})$ as coordinates $\xi$ on a graded manifold, then this Poisson bracket is associated to the symplectic form $\Omega=(d\xi,d\xi)$.

The resulting (shifted) differential graded Lie algebra is formal, and the Poisson bracket induces the vanishing bracket on the cohomology. I have written down a proof, which uses explicit generators for the cohomology (essentially, the Chern–Simons classes). Is there a published proof of this result, perhaps less computational?

After posting this question, I learned that this construction is discussed in the article

Shifted Poisson and symplectic structures on derived N-stacks Jon Pridham (Examples 3.31)

Unless I am mistaken, the special case where $\mathfrak{g}$ is semisimple is not addressed in Pridham's article. I am grateful for all of the general bibliographic references, but I am interested in a very specific result, and none of the comments below address the question I asked.

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    $\begingroup$ Something doesn’t smell right. What if g is abelian? Then the differential is zero, so it is automatically formal, but the bracket is not zero. $\endgroup$ Oct 13, 2021 at 21:24
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    $\begingroup$ Those co-ordinates are in the wrong degree to give a derived affine scheme. That's a form of Lie algebroid, enhancing things in the opposite direction, so $H^*$-isomorphism is too weak to give an equivalence on the associated stacks. $\endgroup$ Oct 13, 2021 at 21:27
  • $\begingroup$ Isn’t this the standard 2-shifted symplectic structure on BG,(associated to an invariant inner product) pulled back to its formal completion (B exp(g) )? $\endgroup$ Oct 13, 2021 at 22:58
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    $\begingroup$ @DavidBen-Zvi : yes, as in Examples 3.31 of arxiv.org/abs/1504.01940 , for instance. $\endgroup$ Oct 13, 2021 at 23:14
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    $\begingroup$ The paper of Pridham cited is a terrific reference for a very general construction including this one - but the explicit example in the question (for reductive groups) is already in the introduction of the original Pantev-Toën-Vaquié-Vezzosi paper on shifted symplectic structures (they say it for the group, but here we're just taking the formal completion of their example). $\endgroup$ Oct 14, 2021 at 15:41

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I still hope that someone will answer this question. In the meantime, I will post a sketch of the proof that I found. It uses Chevalley's theorem, that the space of invariant polynomials $I(\mathfrak{g})$ of a reductive Lie algebra is spanned by the trace polynomials $\text{Tr}_V(\rho(x)^k)$, where $\rho:\mathfrak{g}\to V$ is a finite dimensional representation of $\mathfrak{g}$. This implies 2 things: 1) for any invariant polynomial, $P(\frac12[\theta,\theta])$ vanishes, where $\theta\in C^1(\mathfrak{g},\mathfrak{g})$ is the Maurer-Cartan form; and 2) the algebra $C^*(\mathfrak{g})$ is generated by derivatives $(\theta,\nabla P(\frac12[\theta,\theta]))$. The vanishing of the Poisson bracket on $C^*(\mathfrak{g})^{\mathfrak{g}}$ follows.

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